// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_CONJUGATE_GRADIENT_H
#define EIGEN_CONJUGATE_GRADIENT_H

namespace Eigen {

namespace internal {

/** \internal Low-level conjugate gradient algorithm
 * \param mat The matrix A
 * \param rhs The right hand side vector b
 * \param x On input and initial solution, on output the computed solution.
 * \param precond A preconditioner being able to efficiently solve for an
 *                approximation of Ax=b (regardless of b)
 * \param iters On input the max number of iteration, on output the number of performed iterations.
 * \param tol_error On input the tolerance error, on output an estimation of the relative error.
 */
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
EIGEN_DONT_INLINE void
conjugate_gradient(const MatrixType& mat,
				   const Rhs& rhs,
				   Dest& x,
				   const Preconditioner& precond,
				   Index& iters,
				   typename Dest::RealScalar& tol_error)
{
	using std::abs;
	using std::sqrt;
	typedef typename Dest::RealScalar RealScalar;
	typedef typename Dest::Scalar Scalar;
	typedef Matrix<Scalar, Dynamic, 1> VectorType;

	RealScalar tol = tol_error;
	Index maxIters = iters;

	Index n = mat.cols();

	VectorType residual = rhs - mat * x; // initial residual

	RealScalar rhsNorm2 = rhs.squaredNorm();
	if (rhsNorm2 == 0) {
		x.setZero();
		iters = 0;
		tol_error = 0;
		return;
	}
	const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
	RealScalar threshold = numext::maxi(RealScalar(tol * tol * rhsNorm2), considerAsZero);
	RealScalar residualNorm2 = residual.squaredNorm();
	if (residualNorm2 < threshold) {
		iters = 0;
		tol_error = sqrt(residualNorm2 / rhsNorm2);
		return;
	}

	VectorType p(n);
	p = precond.solve(residual); // initial search direction

	VectorType z(n), tmp(n);
	RealScalar absNew = numext::real(residual.dot(p)); // the square of the absolute value of r scaled by invM
	Index i = 0;
	while (i < maxIters) {
		tmp.noalias() = mat * p; // the bottleneck of the algorithm

		Scalar alpha = absNew / p.dot(tmp); // the amount we travel on dir
		x += alpha * p;						// update solution
		residual -= alpha * tmp;			// update residual

		residualNorm2 = residual.squaredNorm();
		if (residualNorm2 < threshold)
			break;

		z = precond.solve(residual); // approximately solve for "A z = residual"

		RealScalar absOld = absNew;
		absNew = numext::real(residual.dot(z)); // update the absolute value of r
		RealScalar beta = absNew / absOld; // calculate the Gram-Schmidt value used to create the new search direction
		p = z + beta * p;				   // update search direction
		i++;
	}
	tol_error = sqrt(residualNorm2 / rhsNorm2);
	iters = i;
}

}

template<typename _MatrixType,
		 int _UpLo = Lower,
		 typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar>>
class ConjugateGradient;

namespace internal {

template<typename _MatrixType, int _UpLo, typename _Preconditioner>
struct traits<ConjugateGradient<_MatrixType, _UpLo, _Preconditioner>>
{
	typedef _MatrixType MatrixType;
	typedef _Preconditioner Preconditioner;
};

}

/** \ingroup IterativeLinearSolvers_Module
  * \brief A conjugate gradient solver for sparse (or dense) self-adjoint problems
  *
  * This class allows to solve for A.x = b linear problems using an iterative conjugate gradient algorithm.
  * The matrix A must be selfadjoint. The matrix A and the vectors x and b can be either dense or sparse.
  *
  * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
  * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
  *               \c Upper, or \c Lower|Upper in which the full matrix entries will be considered.
  *               Default is \c Lower, best performance is \c Lower|Upper.
  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
  *
  * \implsparsesolverconcept
  *
  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
  * and NumTraits<Scalar>::epsilon() for the tolerance.
  *
  * The tolerance corresponds to the relative residual error: |Ax-b|/|b|
  *
  * \b Performance: Even though the default value of \c _UpLo is \c Lower, significantly higher performance is
  * achieved when using a complete matrix and \b Lower|Upper as the \a _UpLo template parameter. Moreover, in this
  * case multi-threading can be exploited if the user code is compiled with OpenMP enabled.
  * See \ref TopicMultiThreading for details.
  *
  * This class can be used as the direct solver classes. Here is a typical usage example:
	\code
	int n = 10000;
	VectorXd x(n), b(n);
	SparseMatrix<double> A(n,n);
	// fill A and b
	ConjugateGradient<SparseMatrix<double>, Lower|Upper> cg;
	cg.compute(A);
	x = cg.solve(b);
	std::cout << "#iterations:     " << cg.iterations() << std::endl;
	std::cout << "estimated error: " << cg.error()      << std::endl;
	// update b, and solve again
	x = cg.solve(b);
	\endcode
  *
  * By default the iterations start with x=0 as an initial guess of the solution.
  * One can control the start using the solveWithGuess() method.
  *
  * ConjugateGradient can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example
  \endlink.
  *
  * \sa class LeastSquaresConjugateGradient, class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
  */
template<typename _MatrixType, int _UpLo, typename _Preconditioner>
class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType, _UpLo, _Preconditioner>>
{
	typedef IterativeSolverBase<ConjugateGradient> Base;
	using Base::m_error;
	using Base::m_info;
	using Base::m_isInitialized;
	using Base::m_iterations;
	using Base::matrix;

  public:
	typedef _MatrixType MatrixType;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef _Preconditioner Preconditioner;

	enum
	{
		UpLo = _UpLo
	};

  public:
	/** Default constructor. */
	ConjugateGradient()
		: Base()
	{
	}

	/** Initialize the solver with matrix \a A for further \c Ax=b solving.
	 *
	 * This constructor is a shortcut for the default constructor followed
	 * by a call to compute().
	 *
	 * \warning this class stores a reference to the matrix A as well as some
	 * precomputed values that depend on it. Therefore, if \a A is changed
	 * this class becomes invalid. Call compute() to update it with the new
	 * matrix A, or modify a copy of A.
	 */
	template<typename MatrixDerived>
	explicit ConjugateGradient(const EigenBase<MatrixDerived>& A)
		: Base(A.derived())
	{
	}

	~ConjugateGradient() {}

	/** \internal */
	template<typename Rhs, typename Dest>
	void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
	{
		typedef typename Base::MatrixWrapper MatrixWrapper;
		typedef typename Base::ActualMatrixType ActualMatrixType;
		enum
		{
			TransposeInput = (!MatrixWrapper::MatrixFree) && (UpLo == (Lower | Upper)) && (!MatrixType::IsRowMajor) &&
							 (!NumTraits<Scalar>::IsComplex)
		};
		typedef typename internal::conditional<TransposeInput,
											   Transpose<const ActualMatrixType>,
											   ActualMatrixType const&>::type RowMajorWrapper;
		EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree, UpLo == (Lower | Upper)),
							MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
		typedef typename internal::conditional<
			UpLo == (Lower | Upper),
			RowMajorWrapper,
			typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type>::type SelfAdjointWrapper;

		m_iterations = Base::maxIterations();
		m_error = Base::m_tolerance;

		RowMajorWrapper row_mat(matrix());
		internal::conjugate_gradient(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
		m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
	}

  protected:
};

} // end namespace Eigen

#endif // EIGEN_CONJUGATE_GRADIENT_H
